Superconvergence results for linear second-kind Volterra integral equations

Abstract

In this paper, Galerkin method is applied to approximate the solution of Volterra integral equations of second kind with a smooth kernel, using piecewise polynomial bases. We prove that the approximate solutions of the Galerkin method converge to the exact solution with the order O(h r ), whereas the iterated Galerkin solutions converge with the order O(h 2r ) in infinity norm, where h is the norm of the partition and r is the smoothness of the kernel. We also consider the multi-Galerkin method and its iterated version, and we prove that the iterated multi-Galerkin solution converges with the order O(h 3r ) in infinity norm. Numerical examples are given to illustrate the theoretical results.

Two-Grid Methods for Maxwell’s Equations in a Cole-Cole Dispersive Medium

“…refs. [5,7,13,25,37,49], including superconvergence analysis for Maxwell's equations [16,20,28,41,43]. In this paper, we mainly concentrate on the TGM for solving this Cole-Cole model.…”